Appendix AEvaluation of VHM at Load Reference Point The model legs were instrumented with 2 levels of axial gauges A1 & A2 and 3 levels of bending gauges B1 to B3, as shown inFig.. Calib
Trang 1Appendix A
Evaluation of VHM at Load Reference
Point
The model legs were instrumented with 2 levels of axial gauges (A1 & A2) and 3 levels of bending gauges (B1 to B3), as shown inFig A1 Calibration of the gauges and transformation of the measured raw readings (micro-strain) into axial force and bending moment are elaborated inSection 3.3.1.5
Fig A1 Instrumented spudcan leg
The axial force measured is the force required to penetrate the spudcan into soil under undrained condition Either one of the axial forces measured at
Notes:
A denotes axial gauges
B denotes bending gauges
l
l 3
l 2
l 1
Load reference point
Rigid connection
Rigid connection
x
Trang 2A1 & A2 can be used to find V as both give similar values On the other hand, the rotational moment, Mo and horizontal force, Ho at the load reference point (L.R.P.) cannot be measured directly from the experiments but they can be evaluated through extrapolation of the bending moment profiles on the leg as measured by MB1 to MB3at levels B1 to B3
Simple Beam Theory
The model leg has a uniform geometry and constant flexural rigidity, EI With
no consideration of the axial force, simple beam theory can be used to calculate the Ho and Mo at L.R.P as follows:
Take moment about B3
3
Take moment about B2
2
=> (A.2) – (A.1)
……… (A.3)
Substitute (A.3) to (A.1) or (A.2) to calculate M o
2 2
3 3
l H
M
M
or
l H
M
M
o B
o
o B
o
The other bending moment, MB1, can be used to crosscheck the Mo and Ho
obtained above The above equations give sufficiently accurate approximation
2 3
3 2
2 3 3
2
,
) (
l
l
M
M
H
Hence
l l H M
M
B B
o
o B
B
Trang 3the P-∆ effect is negligible When this is not the case, beam-column theory has
to be employed to calculate Mo and Ho
Beam Column Theory:
A beam-column is a member where both bending moment and compressive (axial) force are acting at the same time On some occasions, the deflections may be large enough to add a significant additional moment on the structure The analysis of a beam-column therefore requires consideration of the effect
on the equations of equilibrium of the change of geometry of the bar due to deformation
The governing equation:
)
(x q w
P
w
………… (A.5)
The model leg has a uniform geometry and material property (constant EI).
The general solutions have the following forms:
The integration constants A, B, C and D are to be determined from the boundary conditions of each beam-column
) ( )
(
&
,
) ( sin
cos
x q load lateral the
for solution particular
a
is
x
f
load axial P
EI
P
k
which
in
x f D Cx kx B kx
A
w
) 5 ( )
( ) ( ,
) 5 ( )
( )
cos sin
( ,
) 5 ( )
( )
sin cos
( ,
) 5 (
) ( )
cos sin
( ,
) 5 ( )
( sin
cos
e A PC
x f P x f EI w
P w EI H force
Horizontal
d A x
f EI kx B kx A kP w
EI Q
force
Shear
c A x
f EI kx B kx A P w EI M
Moment
b A x
f C kx B kx A k
w
Slope
a A x
f D Cx kx B kx A w
Deflection
Trang 4For all cases studied in this thesis, q(x) = 0 and the axial load, P is V that were
measured by the axial gauges (see Fig A1)
……… (A.7) Consider the above two boundary conditions, (A.5c) and (A.5e) become:
……… (A.8)
……… (A.9)
………… (A.10)
……… (A.11)
… (A.12)
(A.13)
0
)
0
(
:
2
0
;
0
)
0
(
:
1
:
C
Bk
x
w
connection rigid
at nt displaceme rotational
No
D
A
x
w
connection rigid
at deflection
No
conditions
Boundary
k
kl H M kl
V
A
kV
H
B
hence V
H
C
V P H
PC
H H l
x
at
M kl B
kl
A
V
M
l
x
M
o o o
o
o
o
o o
sin cos
1
&
) (
;
) sin cos
(
)
(
Vx
H kx kV
H kx
k
kl H M kl V w
Deflection
Hence
M k
kl
H kl
V
A
D
o o
o o
o o
sin 1
cos
sin cos
1 ,
,
sin cos
1
Trang 5Rearrange (A.14)
General forms:
……… (A.15)
……… (A.16)
Substitute any pair of MB1,MB2 and MB3 with the corresponding distance from
the rigid connection, l into equations (A.15) and (A.16) to compute Ho and Mo
(which are horizontal force, H and moment, M acting at L.R.P.)
To calculate the spudcan displacement with respect to L.R.P., the following equation is derived:
k
kl V
H kl k
kl H M V w nt displaceme
o
cos
1 1 sin 1
,
………… (A.17)
As illustrated above, the three major load components (VHM) at L.R.P can be evaluated from the axial and bending strain measurements of the model leg
Derivation of load inclination angle, and normalized eccentricity, e/D
V
H angle
n inclinatio
VD
M D
e ty eccentrici
kl kl kl
k
H M kl
kl
M
kl
M kl
M kl kl
k
H
a
o a a
o
b
b a
a b
a
o
cos tan
sin cos
cos
cos cos
tan tan